Once upon a time, before there was acrylic, crochet was done with tiny hooks the size of sewing needles with cotton thread. It was very lacy and was used to make decorative collars, lined coin purses and elaborate tablecloths. Here’s an example of filet work by my great grandmother.
Some people still crochet this way, but it’s painstakingly slow and those stitches are tiny. Here’s a coin purse in progress by a friend of mine that I photographed over a year ago. I’m pretty sure it isn’t finished yet.
Things got a little more fun once more colorful dyes became available, but it was still mostly tiny lace. My great aunt made this doily.
Most people imagine the acrylic granny square afghans which were so popular in the 1970s when they think of crochet. There were some ugly, itchy blankets, but I blame the yarn, not the technique. Here’s a lovely granny afghan a friend made me. 
My first crocheted afghan was a collaboration with my mother. She started this project, realized that it was a bigger task than she anticipated and enlisted me to help her finish it. I crocheted half of the squares and together we sewed them together and wove in a ridiculous number of ends. It was a massive undertaking even for two people with years of experience. This is a message to all the overenthusiastic new crafters who want to make a king sized monochromatic crochet blanket for their first project. Do yourself a favor and start with a hot pad or a pillow cover.
I think my favorite crochet projects are shawls made in lightweight, drapey fibers. This one is made in a cotton/silk blend and is based on the very popular virus shawl, which is a free pattern. You will need to scroll down a little to upload this pattern.
The next shawl is one of my all time favorites, plus it was fun to make. The yarn is a wool/llama blend, so it’s a cozy party. The pattern is Lost In Time, and yes, it’s another free pattern, because I’m cheap and it’s a great pattern.
This year my challenge was to make a flattering crochet cardigan. Crochet can be thicker and less drapey than knit material. My solution was to do the rabbit hole cardigan with very drapey yarns that have been sitting in my stash. Here it is in a silk/alpaca blend, and you can see how it just drapes like mad. This pattern is basically a giant granny square, but the satiny yarn couldn’t look less like grandma’s afghan.
I tried this pattern again, this time pairing a fuzzy mohair with lace weight alpaca yarn. I crocheted this with a V stitch instead of shell stitch. It’s quite warm in spite of the open, lacy structure. 
So there’s a tour of the world of crochet. In spite of the intimidating examples I’ve shown here, more people agree that it is easier to learn than knitting because you only have to manage one loop at a time instead of the entire row at once. And if you ever see something in a store that you recognize as crochet, remember that somebody made it by hand. There are machines that can knit, there isn’t a machine that can crochet.